MAT 1275: Introduction to Mathematical Analysis
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1 1 MT 1275: Intrdutin t Mtemtil nlysis Dr Rzenlyum Slving Olique Tringles Lw f Sines Olique tringles tringles tt re nt neessry rigt tringles We re ging t slve tem It mens t find its si elements sides nd ngles, given sme f tem First f ll, let s see wt elements must e given Ovius, if nly ngles re given nd n sides, tis inf is nt enug t determine sides sine tringles wit te sme ngles re similr nd my ve differerent sizes S, t lest ne side must e given We nsider ll pssile ses wen ne, tw r tree sides re given s well s sme numer f ngles Mre preisely, fur ses re pssile in slving tringles: 1) One side nd tw ngles re given 2) Tw sides nd n ngle ppsite t ne f tem re given 3) Tw sides nd ngle etween tem re given 4) Tree sides re given Min tls t slve tese prlems re tw imprtnt terems: Lw f Sines nd Lw f Csines Here we nsider Lw f Sines nd te first tw prlems Lw f Sines It is ler tt in ny tringle, te igger side, te igger ppsite ngle Hwever, sides re nt prprtinl t ppsite ngles Fr exmple, in rigt tringle 30 60, if side ppsite t 30 is, ten side ppsite t 60 is 3, wi is nt 2 Lw f Sines sys tt in ny tringle sides re prprtinl t te sines f ppsite ngles In ter wrds, te rti f ny side t te sine f te ppsite ngle remins te sme fr ll tree sides f given tringle Mre frmlly, te fllwing terem is true Terem (Lw f Sines) Cnsider tringle BC: B Ten sin = sin B = sin C C
2 2 Prf Fr simplisity, we nsider nly ute tringle (prf fr tuse tringle is sligtly different, ut similr) Let s drw eigt t te side : B Heigt reks tringle BC int tw rigt tringles: BD nd BCD Frm tringle BD, sin = Terefre, = sin Frm tringle BCD, sin C = Terefre, = sin C Equte te ve expressins fr : sin = sin C Divide t sides f tis equtin y sin sin C nd get = sin sin C Similr rti is true fr te side nd ngle B Te prf is mpleted Lw f Sines wrks perfetly gd fr slving tringles fr te se 1) ve wen side nd tw ngles f tringle re given In tis se tringle is defined uniquely Wit n prlem we n find te tird ngle y sutrting tw given ngles frm 180, nd ten use Lw f Sines t find tw ter sides Exmple 1 Slve tringle, if = 14,! B = 40, nd! C = 75 Slutin We need t find ngle, nd sides nd 1) = 180 B C = = 65 2) Using Lw f Sines, = Frm ere, using ls lultr, we get sin sin B = sin B 14 sin 40! = = 99 sin sin65! 3) gin y Lw f Sines, = Frm ere sin sin C = sinc sin = 14 sin75! =149 sin65! Finl nswer:! = 65, = 99, = 149 D C
3 3 Using Lw f Sines migus Cse We nsider nw te se 2) ve wen tw sides nd n ngle ppsite t ne f tem re given In tis se tringle is nt lwys defined uniquely nd we my fe sme diffiulties t slve it Tis is te migus se We will ssume tt te fllwing dt re given: sides nd, nd ngle ppsite t side Cse: ngle is tuse Tis is simple se sine nly tw ptins re pssile: tringle des nt exist r tringle is unique T understnd wy, let s drw ngle nd mrk side n its slnt side: T get tringle, we need t drw side frm te tp pint t te rizntl side f ngle Osius, if side is t srt, it will nt tu te rizntl side, nd tringle des nt exist: In rder t exist, side must e greter tn Ten tringle is defined uniquely We me up t te fllwing Prpsitin 1 Let tw sides nd, nd tuse ngle ppsite t side re given Ten 1) If, tringle des nt exist 2) If >, tringle exists nd it is unique Nte Prt 1) is ls ler y te fllwing resn: if, ten B ngle is tuse, s B ls must e tuse But tringle nnt ve tw tuse ngles Exmple 2 Slve tringle, if = 18, = 14, nd = 130 Slutin Using Lw f Sines, we ve sin = Frm ere sin B sin 14 sin130 sin B = = =
4 4 Ntie, tt t tis pint we lulted sine f ngle B, ut nt tis ngle itself T restre 1 ngle B frm its sine, we n use te uttn sin n lultr similr t wt we did fr rigt tringles Tis uttn rrespnds t inverse sine We ve B = = sin (0596) 37 Nw it is esy t find ngle C: C = 180 B= = 13 T find side, we n use Lw f Sines gin: = Frm ere, = sinc sin sin C sin = 18sin13! = 53 sin130! Finl nswer: B= 37, C= 13, = 53 Cse: ngle is ute s fr tuse ngle, let s drw ngle nd mrk side n its slnt side: T rete tringle, we drw side frm te tp pint Here fur ses re pssile: 1) Side is t srt t meet wit te rizntl side: Tringle des nt exist 2) Side tues rizntl side extly in ne pint: We ve rigt tringle wi is unique 3) Side intersets rizntl side in tw pints: We ve tw tringles wit sides, nd ngle : ne is ute nd te ter is tuse
5 5 4) Side is lng enug nd t ret tringle, side intersets rizntl side nly in ne pint: Te tringle is unique Te tp ngle my e ute r tuse Hw n we distinguis tese fur ses using te vlues f sides, nd ngle? Tke lk t tis piture In yur mind, drw side frm te tp pint Yu n see tt if <, side is t srt nd tringle des nt exist If =, we n drw nly ne rigt tringle If < <, side n e drwn n t sides (left nd rigt) f te eigt, nd we ve tw tringles Finlly, if, we n drw nly ne tringle Ntie tt = sin, s sin = We me up t te fllwing Prpsitin 2 Let tw sides nd, nd ute ngle ppsite t side re given 1) If, tringle is unique Tis tringle my e ute r tuse 2) If <, dente = sin ) If <, tringle des nt exist ) If =, tringle is unique Tis tringle is rigt ) If >, tere re tw tringles One f tem is ute, te ter is tuse Prtil wy t use Prpsitin 2 is t diretly pply Lw f Sines sin slve tis equtin fr sin B : sin B = Tree ses re pssile ere: 1) sin B > 1 Beuse sin B nnt e greter tn 1, tringle des nt exist 2) sin B = 1 We ve B = = sin (1) 90 sin = nd sin B Te tringle is unique It is rigt tringle 3) sin B <1 Let sin B = s, ten B = sin ( s) ngle B (s inverse sine f psitive vlue) is lwys psitive nd ute S, ne tringle lredy exists T understnd weter nter tringle exists, ntie tt tere is ne mre ngle wit te sme sine
6 6 s fr ngle B: suplementl ngle Bʹ = 180 B ngle Bʹ is tuse Suld we ept it s send slutin r rejet it? Just mpre nd nd use te ide tt te igger side, te igger ppsite ngle ) If, ten B ʹ But ngle Bʹ is tuse nd nnt e equl t r less tn ute ngle, s send tringle des nt exist ) If <, te send tringle exists ving te tuse ngle Bʹ = 180 B Nte nter wy t see weter nter tringle exists, is t lulte suplementl ngle Bʹ = 180 B in ny se (regdless n wi side is igger: r ) Ten, if B ʹ+ <180!, ept B ʹ, nd if B ʹ+ 180!, rejet it (n rm fr ngle C) Exmple 3 Let = 20 nd = 30 Determine te numer f tringles tt stisfy te given nditins If tringle exists, slve it 1) = 5 2) = 3) = 16 4) = 25 sin Slutin Using Lw f Sines =, we ve sin B = Frm lultr sin sin B (r just ntie tt 30 is speil ngle), sin = sin30 = 05, nd expressin fr sin B emes sin B = = 1) If = 5, ten sin B = = 2 Beuse sine nnt e greter tn 1, tringle des nt 5 exist 2) If =, ten sin B = = 1 nd B = sin (1) = 90 Tis is rigt tringle T slve it, it remins t lulte ngle C nd side C = 90 B= = 60 Side n e fund y Pytegren Terem (ntiett is yptenuse, nd nd re legs): = = 20 = 300 = 3 Finl nswer: B= 90, C = 60, = 3 3) If = 16, ten sin B = = 0625 nd B = sin (0625) = 39 nter ngle Bʹ, 16 su tt sin Bʹ = sin B is n tuse ngle We ept it euse > ngle Bʹ is suplement t ngle B: Bʹ = 180 B= = 141
7 7 S, we ve tw tringles Let s slve tem It remins t find ngle C nd side ) Tringle wit ngle B = 39 We ve C = 180 B= = 111 By Lw f Sines, = Frm ere, sin sin C sinc 16sin111 = = = 2087 sin sin 30 ) Tringle wit ngle B = 141 (we use letter B insted f Bʹ)We ve C = 180 B= = 9 By Lw f Sines, = sin sin Cʹ Frm ere, sinc 16sin9 = = = 501 sin sin 30 Finl nswer: Tere re tw tringles: B= 39, C = 111, = 2087 B= 141, C = 9, = 501 4) If = 25, ten sin B = = 04 nd B = sin (04) = 24 nter ngle Bʹ, su 25 tt sin Bʹ = sin B is suplement t B nd it is tuse ngle We rejet it euse < nd ngle Bʹ nnt e tuse s we mentined in Nte ve, we n ls lulte B ʹ =180! B =180! 24! =156! Ten B ʹ+ =156! + 30! =186! >180! (nd n rm remins fr ngle C) Terefre, gin we rejet Bʹ S, we ve nly ne tringle wit B = 24 T slve it, it remins t find ngle C nd side C = 180 B= = 126 By Lw f Sines, = Frm ere, sin sin C Finl nswer: B= 24, C = 126, = 4045 sin C 25sin126 = = = 4045 sin sin 30
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